View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 16 (1983) l-9 1 North-Holland ON PROJECTIONS AND LIMIT MAPPINGS OF INVERSE SYSTEMS OF COMPACT SPACES J.J. CHARATONIK and W.J. CHARATONIK Institute of Mathematics, Unicersity of Wrocfaw, Poland Received 11 November 1981 Revised 30 July 1982 For some classes K of mappings we discuss two problems connected with limits of inverse systems: (1) Does the condition that all bonding mappings are in K imply that all projections are in K? (2) Does the condition that all mappings between factor spaces of two given inverse systems are in K imply that the limit mapping between the inverse limit spaces is in K? We answer both these questions in the affirmative for the classes of monotone, of confluent and of weakly confluent mappings of compact spaces, and for some generalizations of these mappings. AMS (MOS) Subj. Class. (1980): Primary 54B25, 54ClO inverse system projection limit mapping monotone confluent weakly confluent Introduction Let S = {X”, f”“, A} and T = {Y”, g”, 2) be inverse systems and let {cp,h”} be a mapping of S into T. For some classes K of mappings the problems when h” E K implies $n{q, h”} E K and when f *- EK implies x* E K were discussed by a number of authors (see e.g. Cl, 4, 81). In the present note we discuss the same problems for more special classes K of mappings, namely for mappings which are monotone, confluent or weakly confluent at some points of their ranges or relative to some points of their domains; as corollaries we get the corresponding results for monotone, for confluent and for weakly confluent mappings. Preliminaries Topological spaces considered throughout this paper are assumed to be compact (thus Hausdorff, see [3, p. 1651) and the mappings are continuous. By a continuum we mean a compact connected space. C(X) denotes the hyperspace of subcontinua of X with the Vietoris topology [3, p. 1621. Given a mapping f:X+ Y, we define f*: C(X) + C(Y) by f,(K) = f (K) if K E C(X). 0166-8641/83/$03.00 @ 1983 Elsevier Science Publishers B.V. (North-Holland) 2 J.J. Charatonik, W.J. Chararonik/Projections and limit mappings A mapping f:X + Y from a topological space X onto a topological space Y is said to be (a) monotone, provided that the inverse image of each subcontinuum of Y is connected; (b) confluent, if for each subcontinuum Q of Y each component of f-‘(Q) is mapped onto the whole Q [2, p. 2131; (c) weakly confluent, if for each subcontinuum Q of Y there exists a component of f-‘(Q) which is mapped onto the whole Q [5, p. 981. Mackowiak [6, p. 7201 has generalized the concept of a monotone mapping saying that a mapping f:X+ Y is monotone relative to a point p EX if for each subcon- tinuum Q of Y such that f(p) E Q the inverse image f-‘(Q) is connected. Recall [6, Theorem 2.1, p. 7201 that: (1) A mapping is monotone if and only if it is monotone relative to each point of its domain. We introduce the following three concepts, which generalize the notion of a confluent mapping. A mappingf:X + Y from a topological space X onto a topologi- cal space Y is said to be (a) cOltfluent at a point q E Y, if for each subcontinuum Q of Y such that 4 E Q each component of f-‘(Q) is mapped onto the whole Q under f; (b) co@uent relative to a point p E X, if for each subcontinuum Q of Y such that f(p) E Q the component of f-‘(Q) containing the point p is mapped onto the whole Q under f; (c) weakly confluent at a point q E Y, if for each subcontinuum Q of Y such that q E Q there exists a component of f-‘(Q) which is mapped onto the whole Q under f- Note that the following statements are immediate consequences of the above definitions. (2) Each mapping f monotone relative to a point p is confluent relative to p, and confluent at f(p). (3) Each mapping f confluent at a point f(p) is confluent relative to p, and each mapping f confluent relative to a point p is weakly confluent at f(p). (4) A mapping f is confluent at a point 4 E Y if and only if it is confluent relative to each point of f-l(q). (5) A mapping is confluent (weakly confluent) if and only if it is confluent (weakly confluent, respectively) at each point of its range. (6) A mapping is confluent if and only if it is confluent relative to each point of its domain. The following notation will be used. {X”, f”“, A} denotes an inverse system of topological spaces X” with continuous bonding mappings fArr :Xlr *X” for any A CCL, where A, p ELI, and A is a set directed by the relation 6. We assume that f *’ is the identity, and we denote by X = l&(X”, f*“, A} the inverse limit space. Further, T* :X+X” denotes the projection from the inverse limit space into the J-I. Chararonik, WJ. Charatonik/Projections and limit mappings 3 Ath factor space. Given a point p EX = @{X^, fhrr, A}, we put p* = r”(p) EX* and we write p = {p”}. Obviously we have (7) f”“(p&) =p* for any A, cc EA with A s CL. A point p EX, i.e., a system of points p* E X” for A E A satisfying (7) is called a thread. Suppose we are given two inverse systems S = {X”, f*‘, A} and T = {Y”, g”, 2). By a mapping of S to T we mean a family {q, h”} consisting of a nondecreasing function cp: Z + A such that the set cp(1) is cofinal in A, and of continuous mappings h”:X*(“)+ Y”, defined for all u E_T and such that g”h’ = hmffcp(o)o(r),i.e., such that the diagram o,o,r,s, f X@(T) X w(o) h’ I is commutative for any U, 7 EZ satisfying u s T. Any mapping of S to T induces a continuous mapping of X = I& S to Y = & T, called the limit mapping induced by {cp,h”} and denoted by h = !im {a, h”}; X + Y (see [3, p. 138 and 1391). PART A: PROJECTIONS In this part we discuss the following problem. Given a class K of mappings, does the condition that all bonding mappings f”’ are in K imply that all projections P” are in K, too? We show an affirmative answer to this question in case when K is the class of monotone, confluent and weakly confluent mappings at some point of the range or relative to some point of the domain. Moreover, a slightly stronger result is obtained: for some fixed element a of the directed set, the condition p” E K for all A with (Yc A implies a’ E K. As corollaries we get corresponding theorems for monotone, confluent and weakly confluent mappings. In what follows A always denotes a directed set and a! is an arbitrary but fixed element of A. Further, in the whole Part A, we consider an inverse system {X”, f”“, A} of compact spaces X’ and we denote by X its inverse limit. Since the set {A CA: a <A} is cofinal in A, we can assume by [3, 2.5.11, p. 1401 without loss of generality that (Yd A for all A E A. We start with the following lemma. Lemma 1. Let a be the only first element of a directed set A and let X = b{X”, f*“, A}. For Q c X” put Q* = (7” J-‘(Q). Then (9) W)-‘(Q) =&lQ”,f”’ IQ”, A). 4 J.J. Charatonik, W.J. CharatoniklProjecrions and limit mappings Proof. Observe that f*“(Q”) c Q”, whence the considered inverse system in the right member of (9) is well defined. Now both inclusions which form equality (9) are direct consequences of the definition of the inverse limit. 1. Monotone mappings The main result of this section is: Theorem 1. If there exists a thread p = {p^} such that for each A E A with a s A the bonding mapping PA is monotone relative to p*, then the projection rTe is monotone relative to p. Proof. Take a continuum Q in X” which contains the point p” = x”(p). We have to show that the set (*“)-l(Q) is connected. To this effect put Q* = (p*)-‘(Q) for A E A. Note that pA E Q”, and since the mapping PA is monotone relative to the point p*, we conclude that Q* is a continuum for each A EA. Therefore (rr”)-r(Q) is the inverse limit of continua 0” according to (9) of Lemma 1, so it is a continuum by [3, Theorem 6.1.18, p. 4361. Corollary 1. If there exists a thread p = {p^} such that the bonding mappings f^“ are monotone relative to pp for all A, u E A with A CCL, then the projections T* are monotone relative to p for all A E A. From Corollary 1 and (1) we get (see [l, Lemma 4.2, p. 2411). Corollary 2. If all bonding mappings f”” are monotone, then all projections T’ are monotone, too.